A 6.2% annual rise in prices in September takes average house values to £170,733 and represents the biggest year-on-year increase since 2010.
The latest increase follows a 5.4% annual rise in August, which like September's increase was the strongest rate since June 2010, the lender said.
Fears have been growing that some stronger than expected price rises this year could be leading to a bubble south of the Border, with borrowers over-stretching themselves.
In Scotland, the Government's Help to Buy scheme is likely to drive prices up by providing loans for up to 20% of the purchase price of a new property for struggling first-time buyers and so-called second-steppers moving home.
Martin Ellis, housing economist at Halifax, said: "There are signs that supply is beginning to respond to the pick-up in demand, which if continued should help to constrain the upward pressure on prices.
"Levels of housebuilding are also increasing, albeit from a very low base."